Smoothing and the Timing of Participating Benefits

Why Timing Matters in Participating Benefits

A participating policy may provide guaranteed benefits and discretionary benefits. The guaranteed amounts follow the contract terms; discretionary amounts depend on declarations or a specified projection rule. Even when a smoothing rule preserves the total amount allocated over a projection, it can change when that amount is paid. Since payments at different durations have different present values, changing their timing can change a projected policy value without changing any guarantee.

This tutorial studies a particular, explicit rule: a fixed fraction of each projected discretionary amount is deferred by one year. It is a useful way to isolate timing effects. The results are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $q_t$ be the discretionary benefit allocated to duration $t$ before smoothing, for $t=1,\ldots,n$. Under a one-period carry-forward smoothing rule with parameter $\lambda$, where $0\leq\lambda\leq1$, the amount paid at duration $t$ is $$ b_1=(1-\lambda)q_1,\qquad b_t=(1-\lambda)q_t+\lambda q_{t-1}\quad (2\leq t\leq n),\qquad b_{n+1}=\lambda q_n. $$ Thus, $1-\lambda$ of each $q_t$ is paid at duration $t$, and $\lambda$ is paid one year later. The final payment at duration $n+1$ is the deferred part of $q_n$.

This rule is defined for a projected sequence of amounts. It does not assert that an insurer must use this rule, or that all participating policies are smoothed in this way. In an actual projection, the smoothing convention, declaration basis, benefit conditions, and treatment of policy exits must be specified. In particular, a bonus already declared and vested under the contract is not made discretionary again by applying a projection rule to future benefits.

For the examples below, the sequence $q_t$ is deterministic and stated per policy. Unless an example says otherwise, the policy is assumed to remain in force throughout the projection, so survival probability is 1 at every payment date. There is no stochastic distribution for the projected amounts. Interest is effective annually, and all payments occur at the end of the stated duration.

Applying the Rule and Checking the Boundary

The parameter $\lambda$ determines how much of each amount moves to the next duration. If $\lambda=0$, there is no smoothing and $b_t=q_t$ for $t=1,\ldots,n$. If $\lambda=1$, every amount is paid one year later. Intermediate values split each amount between its original duration and the following duration. The rule includes a final payment at $n+1$ so the deferred portion of the last projected amount is not lost.

Worked Example: Allocating Benefits Across Durations

A hypothetical participating policy has pre-smoothing discretionary amounts of $q_1=\$1{,}200$, $q_2=\$0$, and $q_3=\$800$. Apply the rule with $\lambda=0.25$. The annual effective interest rate is $i=4\%$; there is no mortality or lapse during the projection, so survival probability is 1 at every duration. The amounts are deterministic; no probability distribution is assumed. Guaranteed benefits are $\$500$ at each of durations 1, 2, and 3.

Duration 1. Pay 75% of the first discretionary amount: $$ b_1=(1-0.25)(1200)=0.75(1200)=\$900. $$

Duration 2. The current amount is zero, but the deferred part of $q_1$ arrives: $$ b_2=0.75(0)+0.25(1200)=\$300. $$

Duration 3. Pay 75% of $q_3$, plus the deferred part of $q_2$, which is zero: $$ b_3=0.75(800)+0.25(0)=\$600. $$

Duration 4. Pay the final deferred part: $$ b_4=0.25(800)=\$200. $$

The smoothed discretionary amounts total $\$900+\$300+\$600+\$200=\$2{,}000$, equal to the original $\$1{,}200+\$0+\$800=\$2{,}000$. The guaranteed amounts remain $\$500$ at durations 1, 2, and 3; they are not multiplied by $1-\lambda$ or moved to other durations. The method reallocates the discretionary stream and avoids the mistake of smoothing the guaranteed benefits as well.

The conservation check in the example follows from the rule, not from the particular numbers. Each $q_t$ contributes $(1-\lambda)q_t$ at duration $t$ and $\lambda q_t$ at duration $t+1$. Those two parts sum to $q_t$. The final duration is essential: omitting $b_{n+1}$ would omit $\lambda q_n$ from the projected total.

Result: On a path where the policy remains in force, the rule preserves the undiscounted total: $$ \sum_{t=1}^{n+1}b_t=\sum_{t=1}^{n}q_t. $$ This is a statement about allocated amounts across durations. It does not imply that their present values are equal, or that aggregate expected payments remain equal when policy exits affect eligibility for later payments.

Why Smoothing Changes Present Value

Let $v=1/(1+i)$, where $i$ is the effective annual interest rate. If each payment is certain and made at the end of its duration, the present value at time 0 of the unsmoothed discretionary stream is $\sum_{t=1}^{n}q_tv^t$. Under the rule, the same original amount $q_t$ is divided between times $t$ and $t+1$. Its present value is therefore $(1-\lambda)q_tv^t+\lambda q_tv^{t+1}$.

Summing this contribution over all original amounts gives a direct derivation of the smoothing effect:

\begin{aligned} \operatorname{EPV}_{\mathrm{smooth}} &=\sum_{t=1}^{n}\left((1-\lambda)q_tv^t+\lambda q_tv^{t+1}\right)\\ &=\left((1-\lambda)+\lambda v\right)\sum_{t=1}^{n}q_tv^t\\ &=\left(1-\lambda(1-v)\right)\operatorname{EPV}_{\mathrm{unsmoothed}}. \end{aligned}

The factorization holds because the same fraction $\lambda$ of every original amount is deferred by exactly one year, with no survival variation or other payment condition. For positive $i$, $v<1$, so deferring a positive amount reduces its present value. At $i=0$, $v=1$, so shifting payment dates does not change present value. This conclusion is about the discretionary stream under these assumptions, not about a guaranteed benefit that stays at its contractual payment date.

Worked Example: Valuing a Smoothed Stream and Unchanged Guarantees

A hypothetical policy has deterministic discretionary amounts $q_1=\$1{,}000$, $q_2=\$0$, and $q_3=\$2{,}000$. Apply one-period smoothing with $\lambda=0.50$. The effective annual interest rate is $i=5\%$, payments occur at each year-end, and there is no mortality or lapse during the projection. No stochastic distribution is assumed. Guaranteed benefits are $\$400$ at the end of each of years 1, 2, and 3.

First calculate the smoothed discretionary amounts: $$ \begin{aligned} b_1&=0.50(1000)=\$500,\\ b_2&=0.50(0)+0.50(1000)=\$500,\\ b_3&=0.50(2000)+0.50(0)=\$1{,}000,\\ b_4&=0.50(2000)=\$1{,}000. \end{aligned} $$ Their total is $\$3{,}000$, the same as the original discretionary total.

At 5%, $v=1/1.05=0.95238095$, with $v^2=0.90702948$, $v^3=0.86383760$, and $v^4=0.82270247$. The unsmoothed discretionary present value is $$ 1000v+2000v^3 =952.38095+1727.67520 =\$2{,}680.05615. $$

The smoothed discretionary present value is $$ \begin{aligned} 500v+500v^2+1000v^3+1000v^4 &=476.19048+453.51474+863.83760+822.70247\\ &=\$2{,}616.24529. \end{aligned} $$ The decrease is $\$2{,}680.05615-\$2{,}616.24529=\$63.81086$, or $\$63.81$ to the nearest cent. The formula provides a check: $$ (1-0.50(1-v))(2680.05615) =0.97619048(2680.05615) =\$2{,}616.24529. $$

The guaranteed present value is $$ 400(v+v^2+v^3) =400(0.95238095+0.90702948+0.86383760) =\$1{,}089.29921. $$ Consequently, the projected value of these benefits is $\$3{,}769.36$ before smoothing and $\$3{,}705.54$ after smoothing, to the nearest cent. The $\$63.81$ difference comes entirely from the timing of discretionary benefits; the guaranteed benefit value is unchanged.

Projected Values at a Later Duration

A projection may value benefits from a duration $k$, rather than from time 0. If $g_j$ denotes a guaranteed benefit payable at duration $j$, and the total benefit is payable only if the policy is in force, the benefit component of a prospective value is

V_k^{\mathrm{benefits}} =\sum_{j>k}v^{j-k}\,{}_{j-k}p_{x+k}\,(g_j+b_j).

Here ${}_{j-k}p_{x+k}$ is the probability of surviving from duration $k$ to the payment at duration $j$; if survival is not the relevant condition, the appropriate payment-contingency probability must be used instead. This expression is the present value of projected benefits only. A net reserve or other net policy value may also include premiums, expenses, or other cash flows with their appropriate signs and timing.

If a policy exits before the next payment, its deferred portion may not be paid. Therefore, the undiscounted conservation result applies to a continuing policy path. It should not be applied automatically to expected payments across a group when survival, lapse, or other eligibility conditions differ by duration. Those conditions must be reflected in the projected cash flows.

Worked Example: Choosing a Smoothing Fraction for a Value Target

A hypothetical policy has deterministic discretionary amounts $q_1=\$800$ and $q_2=\$1{,}200$. Find the one-period smoothing fraction that reduces their present value by exactly 1% relative to the unsmoothed stream. The effective annual interest rate is $i=4\%$; payments occur at year-end. Assume no mortality or lapse, and no stochastic distribution. Guaranteed benefits are $\$300$ at the end of each of years 1 and 2 and are not smoothed.

Write the smoothed-to-unsmoothed present-value ratio as $1-\lambda(1-v)$. A 1% reduction means this ratio is $0.99$. Since $v=1/1.04=25/26$, solve $$ 1-\lambda\left(1-\frac{25}{26}\right)=0.99, \qquad \lambda\left(\frac{1}{26}\right)=0.01, \qquad \lambda=0.26. $$

The smoothed amounts are $$ \begin{aligned} b_1&=0.74(800)=\$592,\\ b_2&=0.74(1200)+0.26(800)=888+208=\$1{,}096,\\ b_3&=0.26(1200)=\$312. \end{aligned} $$ They total $\$2{,}000$, as do the original amounts. The unsmoothed present value is $$ 800v+1200v^2 =769.230769+1109.467456 =\$1{,}878.698225. $$ The smoothed value is 99% of this: $$ 0.99(1878.698225)=\$1{,}859.911243. $$ The reduction is $\$18.786982$, or $\$18.79$ to the nearest cent. The guarantees remain $\$300$ at durations 1 and 2, with present value $300(v+v^2)=\$565.83$. This method solves for the timing parameter from the present-value target and avoids incorrectly applying the 1% reduction to the guarantees.

Assumptions, Interpretation, and Common Traps

The simple present-value factor is convenient, but it depends on its assumptions. It requires each portion of an original amount to be paid with the same contingency as in the calculation, a constant effective annual interest rate for the one-year shift, and the specified end-of-year timing. If a payment is conditional on survival, and survival to the deferred date is less likely than survival to the original date, the deferred portion has an additional reduction in expected value. If the policy lapses, a contractual rule may instead pay, forfeit, or alter a deferred amount. The benefit terms must determine which projection is appropriate.

At a later valuation duration, only future cash flows belong in the prospective value. A bonus already paid is not a future benefit; a declared vested bonus should be treated according to its contractual status. If a projection includes premiums or expenses, smoothing the discretionary benefit stream does not itself change those other cash flows. To isolate the timing effect, compare otherwise identical projections and change only the discretionary payment dates.

Separate the streams.
List guaranteed benefits and discretionary amounts in distinct columns. Identify which projected amounts are eligible for smoothing.

Apply the stated rule by duration.
Allocate $(1-\lambda)q_t$ to duration $t$ and $\lambda q_t$ to duration $t+1$, including the final deferred payment.

Check amount totals and timing.
For a continuing policy path, verify that the smoothed amounts sum to the original discretionary total. Confirm that guarantees have not been moved.

Value each payment at its own date.
Discount from the valuation date and include survival or other payment-contingency probabilities when the benefit depends on them.

A frequent exam trap is to infer that equal undiscounted totals imply equal policy values. The example with $\lambda=0.50$ shows why that fails: $\$1{,}000$ of discretionary benefit is paid one year later under the rule, and at positive interest its present value is lower. The opposite error is to discount the discretionary stream correctly but then smooth the guaranteed stream too. A reliable calculation labels every cash flow by both type (guaranteed or discretionary) and duration before discounting.

Key takeaway: Under the stated one-period rule, smoothing reallocates each discretionary amount across two payment dates. It preserves the undiscounted total for a continuing policy path, but usually changes present value when interest or payment-contingency probabilities vary by date. Guaranteed benefits remain at their contractual amounts and dates.

Check Your Understanding

For each problem, apply the stated timing and payment assumptions; amounts are per policy.

A hypothetical policy has deterministic discretionary amounts $q_1=\$800$ and $q_2=\$400$, with $\lambda=0.25$. The annual effective interest rate is $i=0\%$, and there is no mortality or lapse. Find the smoothed amounts by duration, including the final deferred payment, and their total. No stochastic distribution is assumed. (Answer: $\$600,\ \$500,\ \$100$; total $\$1{,}200$.)
A hypothetical policy has deterministic amounts $q_1=\$500$, $q_2=\$0$, and $q_3=\$1{,}500$, with $\lambda=0.40$. The annual effective interest rate is $i=5\%$, payments occur at year-end, and there is no mortality or lapse. Find the present value of the smoothed discretionary stream to the nearest cent. No stochastic distribution is assumed. (Answer: $\$1{,}738.20$.)
For a deterministic discretionary stream under one-period smoothing, the annual effective interest rate is $i=5\%$, and the present value is to be reduced by exactly $0.8\%$ from its unsmoothed value. Assume no mortality or lapse and no stochastic distribution. Find $\lambda$. (Answer: $\lambda=0.168$.)
A hypothetical policy has deterministic amounts $q_1=\$1{,}000$ and $q_2=\$500$, with $\lambda=0.20$. Benefits are paid only if the policy is in force at the payment date. The probabilities of being in force at the ends of years 1, 2, and 3 are $0.90$, $0.80$, and $0.70$, respectively; take $i=0\%$, with no other benefit conditions and no stochastic distribution for the stated amounts. Find the expected value of the smoothed discretionary payments. (Answer: $\$1{,}270.00$.)

What You'll Learn